SIMULATION AND MONTE CARLO Some General Principles

1. SIMULATION AND MONTE CARLOSome General Principles James C. Spall Johns Hopkins University Applied Physics Laboratory September 2007

2. Overview • Basic principles • Advantages/disadvantages • Classification of simulation models • Role of sponsor in simulation study • Verification, validation, and accreditation • Parallel and distributed computing • Example of Monte Carlo in computing integral • What course will/will not cover • Homework exercises • Selected references

3. Basics • System: The physical process of interest • Model: Mathematical representation of the system • Models are a fundamental tool of science, engineering, business, etc. • Abstraction of reality • Models always have limits of credibility • Simulation:A type of model where the computer is used to imitate the behavior of the system • Monte Carlo simulation: Simulation that makes use of internally generated (pseudo) random numbers

4. Ways to Study System System Experiment w/ actual system Experiment w/ model of system Physical Model Mathematical Model Analytical Model Simulation Model Focus of class

5. Some Advantages of Simulation • Often the only type of model possible for complex systems • Analytical models frequently infeasible • Process of building simulation can clarify understanding of real system • Sometimes more useful than actual application of final simulation • Allows for sensitivity analysis and optimization of real system without need to operate real system • Can maintain better control over experimental conditionsthan real system • Time compression/expansion: Can evaluate system on slower or faster time scale than real system

6. Some Disadvantages of Simulation • May be very expensive and time consuming to build simulation • Easy to misuse simulation by “stretching” it beyond the limits of credibility • Problem especially apparent when using commercial simulation packages due to ease of use and lack of familiarity with underlying assumptions and restrictions • Slick graphics, animation, tables, etc. may tempt user to assign unwarranted credibility to output • Monte Carlo simulation usually requires several (perhaps many) runs at given input values • Contrast: analytical solution provides exact values

7. Classification of Simulation Models • Static vs. dynamic • Static: E.g., Simulation solution to integral • Dynamic: Systems that evolve over time; simulation of traffic system over morning or evening rush period • Deterministic vs. stochastic • Deterministic:No randomness; solution of complex differential equation in aerodynamics • Stochastic (Monte Carlo): Operations of store with randomly modeled arrivals (customers) and purchases • Continuous vs. discrete • Continuous: Differential equations; “smooth” motion of object • Discrete:Events occur at discrete times; queuing networks (discrete-event dynamic systems is core subject of books such as Cassandras and Lafortune, 1999, Law, 2007, and Rubinstein and Melamed, 1998)

8. Practical Side: Role of Sponsor and Management in Designing/Executing Simulation Study • Project sponsor (and management) play critical role • Simulation model and/or results of simulation study much more likely to be accepted if sponsor closely involved • Sponsor may reformulate objectives as study proceeds • A great model for the wrong problem is not useful • Sponsor’s knowledge may contribute to validity of model • Important to have sponsor “sign off” on key assumptions • Sponsor: “It’s a good model—I helped develop it.”

9. Verification, Validation, and Accreditation • Verification and validation are critical parts of practical implementation • Verification pertains to whether software correctly implements specified model • Validation pertains to whether the simulation model (perfectly coded) is acceptable representation • Accreditation is an official determination (U.S. DoD) that a simulation is acceptable for particular purpose(s)

10. Relationship of Validation and Verification Error to Overall Estimation Error • Suppose analyst is using simulation to estimate (unknown) mean vector of some process, say  • Simulation output is (say) X; X may be a vector • Let sample mean of several simulation runs be • Value is an estimate of  • Let be an appropriate norm (“size”) of a vector • Error in estimate of  given by:

11. Parallel and Distributed Simulation • Simulation may be of little practical value if each run requires days or weeks • Practical simulations may easily require processing of 109 to 1012events, each event requiring many computations • Parallel and distributed (PAD) computation based on: Execution of large simulation on multiple processors connected through a network • PAD simulation is large activity for researchers and practitioners in parallel computation (e.g., Chap. 12 by Fujimoto in Banks, 1998; Law, 2007, pp. 61–66) • Distributed interactive simulation is closely related area; very popular in defense applications

12. Parallel and Distributed Simulation (cont’d) • Parallel computation sometimes allows for much faster execution • Two general roles for parallelization: • Split supporting roles (random number generation, event coordination, statistical analysis, etc.) • Decompose model into submodels (e.g., overall network into individual queues) • Need to be able to decouple computing tasks • Synchronization important—cause must precede effect! • Decoupling of airports in interconnected air traffic network difficult; may be inappropriate for parallel processing • Certain transaction processing systems (e.g., supermarket checkout, toll booths) easier for parallel processing

13. Parallel and Distributed Simulation (cont’d) • Hardware platforms for implementation vary • Shared vs. distributed memory (all processors can directly access key variables vs. information is exchanged indirectly via “messages”) • Local area network (LAN) or wide area network (WAN) • Speed of light is limitation to rapid processing in WAN • Distributed interactive simulation (DIS) is one common implementation of PAD simulation • DIS very popular in defense applications • Geographically disbursed analysts can interact as in combat situations (LAN or WAN is standard platform) • Sufficiently important that training courses exist for DIS alone (e.g., www.simulation.com/training)

14. Example Use of Simulation: Monte Carlo Integration • Common problem is estimation of where f is a function, x is vector and  is domain of integration • Monte Carlo integration popular for complex f and/or  • Special case: Estimate for scalar x, and limits of integration a, b • One approach: • Let p(u) denote uniform density function over [a, b] • Let Uidenote ith uniform random variable generated by Monte Carlo according to the density p(u) • Then, for “large” n:

15. Integral estimates for varying n n = 20 n = 200 n = 2000 b =  (ans.=2) 2.296 2.069 2.000 b = 2 (ans.=0) 0.847 0.091 0.0054 Numerical Example of Monte Carlo Integration • Suppose interested in • Simple problem with known solution • Considerable variability in quality of solution for varying b • Accuracy of numerical integration sensitive to integrand and domain of integration

16. What Class Will andWill Not Cover • Emphasis is on general principles relevant to simulation • At class end, students will have rich “toolbox,” but will need to bridge gap to specific application • Classwillcover • Fundamental mathematical techniques relevant to simulation • Principles of stochastic (Monte Carlo) simulation • Algorithms for model selection, random number generation, simulation-based optimization, sensitivity analysis, estimation, experimental design, etc. • Classwill notcover • Particular applications in detail • Computer languages/packages relevant to simulation (GPSS, SIMAN, SLAM, SIMSCRIPT, etc.) • Software design; user interfaces; spreadsheet techniques; details of PAD computing; object-oriented simulation • Architecture/interface issues (HLA, virtual reality, etc.)

17. Homework Exercise 1 Suppose a simulation output vector X has 3 components. Suppose that (a) Using the information above and the standard Euclidean (distance) norm, what is a (strictly positive) lower bound to the validation/verification error ? (b) In addition, suppose  = [1 0 1]T and = [2.3 1.8 1.5]T (superscript T denotes transpose). What is ? How does this compare with the lower bound in part (a)? Comment on whether the simulation appears to be a “good” model. Suppose analyst is using simulation to estimate (unknown) mean vector of some process, say  Simulation output is (say) X; X may be a vector Let sample mean of several simulation runs be Value is an estimate of  Let be an appropriate norm (“size”) of a vector Error in estimate of  given by:

18. Homework Exercise 2 This problem uses the Monte Carlo integration technique (see earlier slide) to estimate for varying a, b, and n. Specifically: (a) To at least 3 post-decimal digits of accuracy, what is the true integral value when a = 0, b = 1? a = 0, b = 4? (b) Using n = 20, 200, and 2000, estimate (via Monte Carlo) the integral for the two combinations of a and b in part (a). (c) Comment on the relative accuracy of the two settings. Explain any significant differences.

19. Selected General References in Simulation and Monte Carlo • Arsham, H. (1998), “Techniques for Monte Carlo Optimizing,” Monte Carlo Methods and Applications, vol. 4, pp. 181229. • Banks, J. (ed.) (1998), Handbook of Simulation: Principles, Methodology, Advances, Applications, and Practice, Wiley, New York. • Cassandras, C. G. and Lafortune, S. (1999), Introduction to Discrete Event Systems, Kluwer, Boston. • Fu, M. C. (2002), “Optimization for Simulation: Theory vs. Practice” (with discussion by S. Andradóttir, P. Glynn, and J. P. Kelly), INFORMS Journal on Computing, vol. 14, pp. 192227. • Fu, M. C. and Hu, J.-Q. (1997), Conditional Monte Carlo: Gradient Estimation and Optimization Applications, Kluwer, Boston. • Gosavi, A. (2003), Simulation-Based Optimization: Parametric Optimization Techniques and Reinforcement Learning, Kluwer, Boston. • Law, A. M. (2007), Simulation Modeling and Analysis (4th ed.), McGraw-Hill, New York. • Liu, J. S. (2001), Monte Carlo Strategies in Scientific Computing, Springer-Verlag, New York. • Robert, C. P. and Casella, G. (2004), Monte Carlo Statistical Methods (2nd ed.), Springer-Verlag, New York. • Rubinstein, R. Y. and Melamed, B. (1998), Modern Simulation and Modeling, Wiley, New York. • Spall, J. C. (2003), Introduction to Stochastic Search and Optimization, Wiley, Hoboken, NJ.